Discretized Equations Of Motion Research Articles

Dynamics of non-linear monatomic and diatomic chains at a T=0 first-order phase transition point

One-dimensional monatomic and diatomic chains with harmonic coupling between neighbouring sites and an on-site anharmonic potential V( phi )=C phi 2n+2+B phi n+2+A phi 2+D are examined in the displacive limit, which serves as a model for a structural phase transition. Kink solutions are obtained at the T=0 first-order phase transition point for B2=4AC, A, C>0, B<0 and D=0 for arbitrary values of n. Non-linear periodic solutions (non-linear phonons) are obtained for n=1, and these are also found to be the solutions of discrete equations of motion. The solutions are shown to be stable and have finite energy.

Analytic fixed points of discrete equations of motion: Universality in homogeneous diffusive systems: 484 (M4,I5) P.B. Visscher, Department of Physics and Astronomy, University of Alabama, University, AL 35486, USA.

Analytic fixed points of discrete equations of motion: Universality in homogeneous diffusive systems: 484 (M4,I5) P.B. Visscher, Department of Physics and Astronomy, University of Alabama, University, AL 35486, USA.

Analytic fixed points of discrete equations of motion: Universality in homogeneous diffusive systems

In this paper a method is presented for analytically calculating the fixed points of scalecoarsening (renormalization) transformations which act on equations of motion for diffusive systems. These fixed points demonstrate the universality of Fick's-law diffusive behavior: All microscopic equations of motion, if they have the correct symmetry, satisfy the correct conservation laws (i.e., particle conservation), and are not too pathological, will look the same on a large space and time scale. Study of the eigenfunctions of the transformation shows that the dominant correction on smaller scales is given by the Burnett equation. The method applied to homogeneous systems in this paper is used in an accompanying paper to obtain new exact results for the universal long-time behavior of inhomogeneous systems in arbitrary dimension.

Dynamics of a Nonlinear Diatomic Chain

We examine nonlinear lattice excitations in a model for a solid which may undergo a displacive structural phase transition. In particular, we investigate the field equations for a diatomic chain of atoms including nonlinear potentials on one species. We identify three classes of excitations in this model. These are linearized phonons, large amplitude solitary waves, and nonlinear phonons. Interactions between the linear phonons and the nonlinear modes give rise to a translation mode which is a further solution of the field equations. The linear and nonlinear phonons are also direct solutions of the discrete equations of motion. However, using the molecular dynamics technique we find that kink-type configurations occur for a much larger parameter space in the discrete lattice than they do in the continuum description. All the stable excitations of the field equations should be included in a formal description (e.g. statistical mechanics) of the model.

Critical fixed points in discrete hydrodynamics

A discrete-cell formulation of hydrodynamics was recently introduced, which is exactly renormalizable in a certain sense: if one knows the discrete equations of motion for a certain cell size W and discrete time interval τ, one can accurately numerically calculate the equations of motion on the coarser scales 2 W or 2τ. These coarsening transformations have previously been investigated for the one-dimensional diffusive system. A line of fixed points was found, parameterized by the (positive) diffusivity D'. In this paper we examine the behavior of the coarsening transformation on the D' = 0 manifold in the space of equations of motion for one-dimensional systems. We find another line of fixed points, this one parameterized by the super-Burnett coefficient D' 3. This corresponds to a Gaussian critical point. The possibility of generalizing this to non-Gaussian (Ising-like) critical points is discussed.

A Finite Element Formulation for Fluid-Structure Interaction in Three-Dimensional Space

In this paper a development is presented for a three-dimensional hexahedral hydrodynamic finite-element. Using trilinear shape functions and assuming a constant pressure field in each element, simple relations were obtained for internal nodal forces. Because the formulation was based upon a rate approach it was applicable to problems involving large displacements. This element was incorporated into an existing plate-shell finite element code. Diagonal mass matrices were used and the resulting discrete equations of motion were solved using an explicit temporal integrator. Results for several problems were presented which compare numerical predictions to closed form analytical solutions. In addition, the fluid-structure interaction problem of a fluid-filled, cylindrical vessel containing internal cylinders was studied. The internal cylinders were cantilever supported from the top cover of the vessel and were periodically located circumferentially at a fixed radius. A pressurized cylindrical cavity located at the bottom of the vessel at its centerline provided the loading.

Discrete hydrodynamics: Determination of equations of motion by simulation

A detailed procedure is described for computing discrete equations of motion for a fluid, to implement a new method which turns out to be substantially more efficient than previous methods for calculating transport coefficients. This paper describes the calculations of discrete averages from molecular dynamics data and the numerical extraction of the equation-of-motion coefficients in a way which makes maximum use of the geometric symmetry of the problem. Extrapolation to the infinite-system limit and eventual computation of transport coefficients by renormalization are discussed. The method described in detail here is briefly sketched and applied numerically to computing the viscosity of the soft-sphere liquid in a subsequent paper.

Structures Under Shock And Impact 411 Low Velocity Impact Of A Deformable Multi-body System

An analysis methodology for impact of multi-body systems comprising deformable and rigid bodies is described, and applied to the analysis of a portable radar system which undergoes elastic and plastic deformation as well as significant rigid body motion during impact. A detailed local finite element analysis is first performed for developing simplified non-linear structural properties, followed by a global finite element analysis which includes treatment of rigid bodies through the discrete equations of motion. Certain parameters, such as the flexibility of structural members in the load path, and the characteristics of the impact pad, were found to be essential for a correct simulation.